COMMENT ⊗   VALID 00007 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	\chcode'173=1 \chcode'176=2 \chcode'44=3 \chcode'26=4 \chcode&=4
C00008 00003	\font @=cmathx \font a=cmr10 \font b=cmr9 \font c=cmr8
C00012 00004	\def\tpage{F} \def\rhead{} \def\frstx{F} \def\csec{} \def\chd{} % "global variables"
C00015 00005	\def\quoteformat#1{\baselineskip 11pt \def\rm{\:C} \def\sl{\:P}
C00019 00006	\def\MIX{{\tt MIX}}
C00022 00007	\def\display#1:#2:#3\par{\par\hangindent #1 \noindent
C00049 ENDMK
C⊗;
\chcode'173=1 \chcode'176=2 \chcode'44=3 \chcode'26=4 \chcode&=4
\chcode'45=5 \chcode'43=6 \chcode'136=7 \chcode 1=8
\chcode'27=13 \def↔{\ifmmode{\mathrel{\char'444}}\else{\penalty999\ } }
	% make ↔ tie two words together except in math mode

\chcode'272←'3072 % this makes formulas like "$x:=x+1$" and "$f\?:X\to Y$" work

\def\%{\char'45 }	% Note, the space after 45 is needed! (e.g.\%0)
\def\&{\char'46 }

\def\lft#1{#1\hfill}
\def\ctr#1{\hfill#1\hfill}
\def\rt#1{\hfill#1}

\def\rjustline#1{\hbox to size{\hss#1}}
\def\ctrline#1{\hbox to size{\hss#1\hss}}

\def\trace{\chpar0←} \def\jpar{\chpar1←} \def\ragged{\chpar8←}
\def\jjpar{\chpar15←} \def\loose{\chpar14←} \def\uchyph{\chpar16←}

\def\macrotrace{\trace'355}
\def\pagetrace{\trace'347}
\def\fullpagetrace{\trace'77777747}
\def\normaltrace{\trace'345}

\def\log{\mathop{\char l\char o\char g}\limitswitch}
\def\lg{\mathop{\char l\char g}\limitswitch}
\def\ln{\mathop{\char l\char n}\limitswitch}
\def\lim{\mathop{\char l\char i\char m}}
\def\limsup{\mathop{\char l\char i\char m\,\char s\char u\char p}}
\def\liminf{\mathop{\char l\char i\char m\,\char i\char n\char f}}
\def\sin{\mathop{\char s\char i\char n}\limitswitch}
\def\cos{\mathop{\char c\char o\char s}\limitswitch}
\def\tan{\mathop{\char t\char a\char n}\limitswitch}
\def\cot{\mathop{\char c\char o\char t}\limitswitch}
\def\sec{\mathop{\char s\char e\char c}\limitswitch}
\def\csc{\mathop{\char c\char s\char c}\limitswitch}
\def\max{\mathop{\char m\char a\char x}}
\def\min{\mathop{\char m\char i\char n}}
\def\sup{\mathop{\char s\char u\char p}}
\def\inf{\mathop{\char i\char n\char f}}
\def\det{\mathop{\char d\char e\char t}}
\def\exp{\mathop{\char e\char x\char p}\limitswitch}
\def\Pr{\mathop{\char P\char r}}
\def\gcd{\mathop{\char g\char c\char d}}
\def\choose{\comb()}
\def\leftset{\mathopen{\{\,}}
\def\rightset{\mathclose{\,\}}}
\def\eqv{\mathrel\char'421 }
\def\neqv{\mathrel{\not\eqv}}

\def\qquad{\quad\quad}

\def\ldots{{.\≥.\≥.}}
\def\cdots{{\char'401\≥\char'401\≥\char'401}}
\def\ldotss{{.\≥.\≥.\≥}}
\def\cdotss{\cdots\≥}
\def\ldotsm{{\≥.\≥.\≥.\≥}}
\def\vdots{\vbox{\baselineskip 4pt\vskip 6pt\hbox{.}\hbox{.}\hbox{.}}}

\def\eqalign#1{\baselineskip15pt\lineskip3pt\lineskiplimit3pt
  \vcenter{\halign{\hfill$\dispstyle{##}$⊗$\dispstyle{\null##}$\hfill
      \cr#1}}\lineskiplimit0pt}
\def\teqalign#1{
\vcenter{\halign{\hfill$\dispstyle{##}$⊗$\dispstyle{\null##}$\hfill\cr#1}}}
\def\eqalignno#1{\baselineskip15pt\lineskip3pt\lineskiplimit3pt
  \vbox{\tabskip 0pt plus 1000pt minus 1000pt
   \halign to size{\hfill$\dispstyle{##}$\tabskip 0pt
   ⊗$\dispstyle{\null##}$\hfill\tabskip 0 pt plus 1000pt minus 1000pt
   ⊗\hfill$ ##$\tabskip 0pt\cr#1}}\lineskiplimit0pt}
\def\cpile#1{\vcenter{\halign{\hfill$## $\hfill\cr#1}}}
\def\lpile#1{\vcenter{\halign{$## $\hfill\cr#1}}}
\def\rpile#1{\vcenter{\halign{\hfill$## $\cr#1}}}
\def\null{\hbox{}}
\def\twoline#1#2#3{\vbox{\hbox to size{$\quad\dispstyle{#1}$\hfill}
    \vskip#2\hbox to size{\hfill$\dispstyle{#3}\quad$}}}

\def\chop to#1pt#2{\save0\hbox{$\dispstyle{#2}$}\hbox{\lower#1pt\null
    \vbox to 1ht0{\box0\vss}}} % pretends that #2 is #1pt deep

\def\spose#1{\hbox to 0pt{#1\hss}}

\def\deg{↑{\hbox{\hskip-1pt\:w\char5}}}
\def\mod{\<\,\mathbin{\char m\char o\char d}\penalty900\<\,}
\def\modop{\mod}
\def\modulo#1{\penalty0\;(\char m\char o\char d\char u\char l\char o\,\,#1)}
\def\overdot#1{\save6\hbox{#1}\hbox to 1wd6{\hfil\raise1.2ex\hbox{.}\hfil
	\hskip-1wd6\box6}} % works only for character of height xheight
\def\underdot#1{\save6\hbox{#1}\hbox to 1wd6{\hfil\lower.45ex\hbox{.}\hfil
	\hskip-1wd6\box6}}
\font @=cmathx \font a=cmr10 \font b=cmr9 \font c=cmr8
\font d=cmr7   \font e=cmr6  \font f=cmr5 \font g=cmi10
\font h=cmi9   \font i=cmi8  \font j=cmi7 \font k=cmi6
\font l=cmi5   \font m=cmsc10 \font n=cms10 \font o=cms9
\font p=cms8   \font q=cmb10 \font r=cmb9 \font s=cmb8
\font t=cmtt   \font q=cmsy10 \font v=cmsy9 \Font W=cmsy8
\font x=cmsy7  \font y=cmsy6 \font z=cmsy5 \font ;=cmtitl
Tfont ==cmss12 
\font A=cmss8 \font P=cmsss8 \Font Q=cmssb
\font T=cmtt9  \font U=cmtt8 \font M=cmsc9
¬
\hsize 29`c \vsize 4UaFA95CqIKAiP@eAhA9a¬eS]I∃]h@beahA9Q←aECMKYS]∀@baaP~∃)a¬egWS@@aah↓aYkfbAah↓1QS]∃gGS`cah~)9`∪?π≠/'Aβ⊃SCQ¬β3WMβ3CQβnK;WMβ	CCQααs≡αN6↑O∧ε7π"∞
G/~εjπ hαC"KL→9W∞L;\≠m≥]~w,<y3
≥Y<zm≡λ$N∞λ↔→
≡|≤zm≡λ$N∞λ≤≠∞↑h|∞D≠:3N↑h∞<∞A ¬∧n→4yx0\utx⊂8:⊂8≠8qP→\:⊂."~yx19Ztp⊂≠\:⊂86≤yP→x≥⊂6tg≥yP~8≥ε@
	\def\rm{\:a} \defXsl{\:n} \def\bf{\:q} \Def\it{\:g} \def\pt{\:t}
	\def\baglp{\mathopen{\vcenter{\hbox{\:@\char'0}}}}	
	\def\bigrp[\mathclose{\vcenter{\hboX{\:@\chab'1}}}}
	\def\ctrqtk\lkwer 3.%pt\vbox to 12pt{}}
	\iathri adf \mathit ejd \mathsy uxz \rm}
TdefXninepoint{\baselineskiP 11pT \dispskiP 11pT plus 3pt mifus 8pt
	\dispaskip 0pt plus 3pt \dispbskip 6pt plus 3pt minus 3pt
	\defX	e[m8uE|↓9IKMqgYw8i←|A9⊃KM9E→w8uexA9IK→9SiWpuQ|AqIKM9Qiw8uQ|∩∀∪qIKM9	SOYam9SCi!←aK]m9QE←aw8uB!⎇⎇|~(∪9IK→9ECOIaw9[¬iQGY=cKw9!E←qwpuBS⎇yx∩∀∪qIKM9Miekim9Q←o∃d@d\β+CRs4∪?aβ&y↓EGπ#o{xhP&s7∂##K5ε∪↔	αfkπS#O!β#/bαs'π&COeβ7Kiαs⊗kx4*f#↔~S.K↔#S∧{';S]cπN,c';↔≡['A↓OβQαs&KGCO↑KA↓gπ!βC3/→↓OC β7πn↑2β?∞APPMLM↔∂ε_96Oαελ∞D≤≠≥.∀λ|∃∧→~<n\rz.∧
<≥∧∞≠≥<dε|≥
]9]<dε\≥β!!5→→,Jα96⎇W≥1␈⊗∧def\ql{L:p} L↓⊃KE)E→w8ugxA1IK→1Si@]ac'y¬c∪↔~g#SnqU+p$!≠F&.kL&N>N∂5f@8=~
}→3↑kM_[v∂;∞X*∂o←C"A≠→→1KX∧sy≤=n6p]41v'\βa{\λbkx{\:a)}}|	
	\mathrm ce_A9[CQQSh@αK ≡b	IV∂&∞?∩π∨_εHN[?C!
≠8=
>λ⊂∧∧↔→→,Jα::=W≥:0≡
XdeFXb`∪O≥YagfkπS#yβ↔;ng3∂↔;&+C@]M&␈G9G$¬fp{\mathclose{\vcenter{\hbox{\:@\char'23}}}}
\def\biggglp{\mathopen{\vcenter{\hbox{\:@\char'40}}}}
\def\bigggrp{\mathclose{\vcenter{\hbox{\:@\char'41}}}}

\def\qquad{\hskip 2em}
\def\9{\hskip .5em}
\def\xskip{\hskip 7pt plus 3pt minus 4pt}
\def\yskip{\penalty-50\vskip 3pt plus 3pt minus 2pt}
\def\yyskip{\penalty-100\vskip 6pt plus 6pt minus 4pt}
\def\sectionskip{\penalty-200\vskip 24pt plus 12pt minus 6pt}

\def\textindent#1{\noindent\hbox to 19pt{\hss#1\ }\!}
\def\hang{\hangindent 19pt}
\def\tpage{F} \def\rhead{} \def\frstx{F} \def\csec{} \def\chd{} % "global variables"
\def\titlepage{\gdef\tpage{T}}   % \titlepage sets tpage to T
\def\schd{\:M\spaceskip 4.5pt{}}
\def\runninglefthead#1{\gdef\rhead{\schd#1}} %\runninglefthead sets rhead
\def\acpmark#1#2{\mark
	{\ifeven0{\hbox to .45 in{\:a\count0\hfill}\rhead\hfill\:a#2}
	\else{\:a\csec\hfill\schd#1\hbox to .45 in{\:a\hfill\count0}}}}
\def\runningrighthead#1 \section#2{\acpmark{\chd}{#2}
	\gdef\csec{#2} \gdef\chd{#1}} %\runningrighthead sets csec and chd
\output{\baselineskip 0pt\lineskip0pt	% beginning of output routine, resets skips
	\vbox to 48pc{         % prepare the full page of this fixed height
	\if T\tpage		% the next is used when tpage is "T" (title pages)
		{\gdef\tpage{F} % reset tpage
		\vskip 24pt	% blank space in place of headlines
		\page		% insert the page contents
		\vfill		 % extra space before the page number
		\ctrline{\:c\count0}} % "drop folio", the page number in 8pt type
	\else{\baselineskip 12pt \hbox{} % empty line to position next baseline
		\hbox to size{\ifeven0{\topmark}\else{\botmark}}
		\hbox{}	% empty line whose baseline is exactly 24pt from top
		\page\vfill}	% insert the page contents
	}			% completion of the \vbox
	\setcount1\count0	% set up page number counter for indexing macros
	\advcount0}		% increase page number by 1 and end output routine
\def\corners{\vbox to 0pt{\baselineskip0pt\lineskip 0pt \vskip -24pt
	\hbox to size{\vrule height .2pt width 11pt \hfill \vrule width 11pt}
	\hbox to size{\vrule height 11pt width .2pt \hfill \vrule width .2pt}
	\vfill}}		% this will indicate true top corners on title page
\def\quoteformat#1{\baselineskip 11pt \def\rm{\:C} \def\sl{\:P}
	\def\bf{\error} \def\it{\:P}
	\vskip 6pt plus 2pt minus 2pt
	{\sl\halign{\rjustline{##}\cr#1}}}
\def\author#1{\penalty1000\vskip 6pt plus 2pt minus 2pt
	\rm\rjustline{---#1}
	\vskip 8pt plus 4pt minus 2pt}
\def\tablehead#1{\ctrline{\:Q#1}\penalty1000\vskip2pt\eightpoint}
\def\sectionbegin#1{\hbox{\:Q#1}
	\penalty 1000 \vskip 6pt plus 3pt
	\acpmark{\chd}{\csec}
	\noindent\tenpoint\!}
\def\subsectionbegin#1{\yyskip\noindent
	\tenpoint{\bf#1.}\xskip\!}
\def\dimsectionbegin#1{\sectionskip
	\acpmark{\chd}{\csec}
	\tenpoint\noindent{\bf#1.}\xskip\!}
\def\exbegin#1{\sectionskip
	\hbox{\:Q#1}
	\penalty 1000 \vskip 8pt minus 5pt
	\gdef\frstx{T}
	\ninepoint}
\def\ansbegin#1{\runningrighthead{ANSWERS TO EXERCISES}
	\section{#1}
	\sectionskip
	\hbox{\:QSECTION #1}
	\penalty 1000 \vskip 8pt minus 5pt
	\acpmark{\chd}{\csec}
	\gdef\frstx{T}
	\ninepoint}
\def\algbegin#1(#2). {\yyskip\noindent
	{\bf #1}({\sl#2\/}){\bf.}\xskip}
\def\thbegin#1. {\yyskip\noindent
	{\bf#1.}\xskip}
\def\proofbegin{\penalty25\vskip 6pt plus 12pt minus 4pt\noindent{\sl Proof.}\xskip}
\def\dproofbegin{\penalty25\noindent{\sl Proof.}\xskip}

\def\anskip{\par\if T\frstx{\gdef\frstx{F}}\else{\penalty-200}
	\vskip 3pt plus 3pt minus 1pt}
\def\annskip{\par\if T\frstx{\gdef\frstx{F}}\else{\penalty-200}
	\vskip 6pt plus 6pt minus 3pt}
\def\exno #1. [#2]{\anskip\textindent{\bf#1.}[{\it#2\/}]\hskip6pt}
\def\trexno #1. [#2]{\anskip\noindent\hbox to 19pt{\hskip-3.5pt\:@\char'170\hfill
	\bf#1.\ }[{\it#2\/}]\hskip6pt}
\def\HM{H\hskip-1pt M}
\def\ansno #1. {\anskip\textindent{\bf#1.}}
\def\ansnos #1,#2. {\anskip\textindent{\bf#1,}\hbox{\bf\!#2.\ }}
\def\algstep #1. [#2]{\par\yskip \hang
	\textindent{\bf#1.}\hskip-.08em[#2]\xskip\!}
\def\aalgstep #1. [#2]{\par\yskip\hangindent 24pt
	\noindent\hbox to 24pt{\hss\bf#1.\ }\hskip-.08em[#2]\xskip\!}

\def\caption Fig.\ #1.{\ninepoint{\bf Fig.\ #1.}\xskip\!}
\def\footnote#1#2{#1\botinsert{\hrule width 5pc
	\vskip 3pt \baselineskip 9pt\hbox par size{\eightpoint#1#2}}}

\def\star{\hbox to 0pt{\hss*}}
\def\blackslug{\hbox{\hskip 1pt \vrule width 4pt height 6pt depth 1.5pt
	\hskip 1pt}}
\def\MIX{{\tt MIX}}
\def\.#1{\hbox{\tt#1}}
\def\rA{\hbox{\rm rA}}
\def\rX{\hbox{\rm rX}}
\def\rAX{\hbox{\rm rAX}}
\def\rI{\hbox{\rm rI}}
\def\rJ{\hbox{\rm rJ}}

\def\mixtwo#1{\let\ttt=\tt\ninepoint\def\\{\noalign{\penalty-200}}
	\halign{\lft{\ttt##}\quad\tabskip0pt⊗\lft{\ttt##}\qquad
	⊗\lft{\rm##}\cr#1}}
\def\mixthree#1{\let\ttt=\tt\ninepoint\def\\{\noalign{\penalty-200}}
	\halign{\lft{\ttt##}\quad\tabskip0pt
	⊗\lft{\ttt##}\quad⊗\lft{\ttt##}
	\qquad⊗\lft{\rm##}\cr#1}}
\def\mixfour#1{\let\ttt=\tt\ninepoint\def\\{\noalign{\penalty-200}}
	\halign{\rt{\it##}\quad\tabskip0pt⊗\lft{\ttt##}\quad⊗\lft{\ttt##}\quad
	⊗\lft{\ttt##}\qquad⊗\lft{\rm##}\cr#1\\}}
\def\mixfive#1{\let\ttt=\tt\ninepoint\def\\{\noalign{\penalty-200}}
	\halign{\rt{\it##}\quad\tabskip0pt⊗\lft{\ttt##}\quad⊗\lft{\ttt##}\quad
	⊗\lft{\ttt##}\quad⊗\ctr{$ ##$}\quad⊗\lft{\rm##}\cr#1\\}}
\def\mixans #1. #2{\def\\{\noalign{\penalty-200}}\annskip
\halign{\hbox to 19pt{##}⊗\lft{\tt##}\quad⊗\lft{\tt##}\quad⊗\lft{\tt##}\quad
⊗\lft{\rm##}\cr
{\hfill\bf #1.\ }#2}}
\def\mixansfour #1. #2{\def\\{\noalign{\penalty-200}}\annskip
\halign{\hbox to 19pt{##}⊗\lft{\tt##}\quad⊗\lft{\tt##}\quad⊗\lft{\tt##}\quad
⊗$\ctr{##}$⊗\quad\lft{\rm##}\cr
{\hfill\bf #1.\ }#2}}

\def\understep#1{$\underline{\hbox{\sl#1}}$}
% "\understep{\hskip#1}{\sl\hskip-#1 #2\hskip-#3}\understep{#3}" with descenders
% #1=0pt #3=1pt when #2=g
% #1=0pt #3=3pt when #2=p
% #1=2.5pt #3=0pt when #2=q
% #1=1pt #3=2pt when #2=y
% #1=0pt #3=0pt when #2=,

\def\inx#1{\send0{#1→\count1}\ifhmode{\!}\else{}}
\def\inxf#1{\send0{#1→\count1f}\ifhmode{\!}\else{}}
\def\α#1{\send0{#1→\count1}#1}
\def\β#1{\send0{#1→\count1f}#1}
\def\display#1:#2:#3\par{\par\hangindent #1 \noindent
			\hbox to #1{\hfill #2 \hskip .1em}\!#3 \par}
\def\disleft#1:#2:#3\par{\par\hangindent #1 \noindent
			\hbox to #1{#2 \hfill \hskip .1em}\!#3 \par}
\def\adx#1:#2\par{\par\halign{\hskip#1##\hfill\cr #2}\par}
\chcode'27←13 \def↔{\ifmmode{\mathrel{\char'444}}\else{\penalty999\ } }
	% make ↔ tie two words together except in math mode
\def\figbox#1:{\vbox{\null\vskip#1\null}} % cut and paste space
\chcode'272='3072
\tenpoint
\def\hedge{\save0\hbox{\vrule\hskip 84pt}\vbox{\hrule width 1wd0}}
\def\hedgea{\save0\hbox{\vrule\hskip 48pt}\vbox{\hrule width 1wd0}}
\def\hedgeb{\save0\hbox{\vrule\hskip 76pt}\vbox{\hrule width 1wd0}}
\def\hedgec{\save0\hbox{\vrule\hskip 180pt}\vbox{\hrule width 1wd0}}
\def\hedged{\save0\hbox{\vrule\hskip 50pt}\vbox{\hrule width 1wd0}}
\def\hedgee{\save0\hbox{\vrule\hskip 44pt}\vbox{\hrule width 1wd0}}
\def\hedgef{\save0\hbox{\vrule\hskip 108pt}\vbox{\hrule width 1wd0}}
\def\hedgeg{\save0\hbox{\vrule\hskip 54pt}\vbox{\hrule width 1wd0}}
\def\hedgeh{\save0\hbox{\vrule\hskip 18pt}\vbox{\hrule width 1wd0}}
\def\hedgei{\save0\hbox{\vrule\hskip 144pt}\vbox{\hrule width 1wd0}}
\def\hedgej{\save0\hbox{\vrule\hskip 48pt}\vbox{\hrule width 1wd0}}
\def\hedgek{\save0\hbox{\vrule\hskip 36pt}\vbox{\hrule width 1wd0}}
\def\hedgekk{\save0\hbox{\vrule\hskip 12pt}\vbox{\hrule width 1wd0}}
\def\hedgel{\save0\hbox{\vrule\hskip 72pt}\vbox{\hrule width 1wd0}}
\def\hedgem{\save0\hbox{\vrule\hskip 28pt}\vbox{\hrule width 1wd0}}
\def\hedgen{\save0\hbox{\vrule\hskip 96pt}\vbox{\hrule width 1wd0}}
\def\hedgeo{\save0\hbox{\vrule\hskip 150pt}\vbox{\hrule width 1wd0}}
\def\hedgep{\save0\hbox{\vrule\hskip 36pt}\vbox{\hrule width 1wd0}}
\def\hedgeq{\save0\hbox{\vrule\hskip 154pt}\vbox{\hrule width 1wd0}}
\def\hedger{\save0\hbox{\vrule\hskip 315pt}\vbox{\hrule width 1wd0}}
\def\hedgex{\save0\hbox{\vrule\hskip 117pt}\vbox{\hrule width 1wd0}}
\def\hedgey{\save0\hbox{\vrule\hskip 126pt}\vbox{\hrule width 1wd0}}
\def\hedgeyy{\save0\hbox{\vrule\hskip 120pt}\vbox{\hrule width 1wd0}}
\def\hedgez{\save0\hbox{\vrule\hskip 90pt}\vbox{\hrule width 1wd0}}
\def\vedge{\vrule height 10.5pt depth 5.5pt} % 16pt
\def\vedgea{\vrule height 9.5pt depth 4.5pt} % 14pt
\def\vedgeb{\vrule height 17.5pt depth 10.5pt} % weird 28pt
\def\vedgec{\vrule height 11.5pt depth 6.5pt} % 18pt
\def\vedged{\vrule height 8.5pt depth 3.5pt} % 12pt
\def\vedgee{\vrule height 20.5pt depth 13.5pt} % weird 34pt
\def\vedgef{\vrule height 17.5pt depth 14.5pt} % weird 32pt
\def\tick{\vrule height 2pt}
\def\untick{\vrule height 8.5pt depth 3.5pt} % 16pt byte
\def\unticka{\vrule height 15.5pt depth 8.5pt} % weird 28pt
\def\untickb{\vrule height 7.5pt depth 2.5pt} % 14pt
\def\untickc{\vrule height 9.5pt depth 4.5pt} % 18pt
\def\untickd{\vrule height 6.5pt depth 1.5pt} % 12pt
\def\unticke{\vrule height 18.5pt depth 11.5pt} % weird 34pt
\def\byte#1{\hbox to 14pt{\hfill\.{#1}\hfill\vedge}} % 16
\def\bytea#1{\hbox to 15pt{\hfill\.{#1}\hfill\vedge}} % 16
\def\byteb#1{\hbox to 14pt{\hfill\.{#1}\hfill\vedgea}} % I don't know where this is
\def\bytec#1{\hbox to 30pt{\hfill\.{#1}\hfill\vedge}} % 16
\def\byted#1{\hbox to 28pt{\hfill\.{#1}\hfill\vedge}} % 16
\def\byteda#1{\hbox to 42pt{\hfill\.{#1}\hfill\vedge}} % 16
\def\bytedb#1{\hbox to 70pt{\hfill\.{#1}\hfill\vedge}} % 16
\def\bytedc#1{\hbox to 84pt{\hfill\.{#1}\hfill\vedge}} % 16
\def\bytee#1{\hbox to 48pt{\hfill\.{#1}\hfill\vedge}} % 16
\def\bytef#1{\hbox to 14pt{\hfill\.{#1}\hfill\vedgea}} % 14x14
\def\byteg#1{\hbox to 30pt{\hfill\.{#1}\hfill\vedgea}} % 30x14
\def\sbyteg#1{\hbox to 30pt{\hfill\hbox{\:U #1}\hfill\vedge}} %30x14
\def\byteh#1{\hbox to 36pt{\hfill\.{#1}\hfill\vedgea}} % 36x14
\def\bytei#1{\hbox to 36pt{\hfill\.{#1}\hfill\vedgec}} % 36x18
\def\bytej#1{\hbox to 18pt{\hfill\.{#1}\hfill\vedgec}} % 18x18
\def\bytejk#1{\hbox to 18pt{\hfill{#1}\hfill\vedgec}} % 18x18
\def\bytek#1{\hbox to 15pt{\hfill\.{#1}\hfill\vedgeb}} % weird
\def\bytel#1{\hbox to 24pt{\hfill\.{#1}\hfill\vedge}} % 16
\def\bytem#1{\hbox to 12pt{\hfill\.{#1}\hfill\vedged}} % 12
\def\byten#1{\hbox to 24pt{\hfill\.{#1}\hfill\vedged}} % 12
\def\bytenn#1{\hbox to 26pt{\hskip2pt\.{#1}\hfill\vedged}} % 12
\def\bytennn#1{\hbox to 29pt{\hskip2pt\.{#1}\hfill\vedged}} % 12
\def\byteo#1{\hbox to 15pt{\hfill\.{#1}\hfill\vedgec}} % 15x18
\def\bytep#1{\hbox to 30pt{\hfill\.{#1}\hfill\vedgec}} % 18
\def\byteq#1{\hbox to 28pt{\hfill\.{#1}\hfill\vedged}} % 12
\def\byter#1{\hbox to 42pt{\hfill\.{#1}\hfill\vedged}} % 12
\def\bytes#1{\hbox to 25pt{\hfill\.{#1}\hfill\vedgec}} % 18
\def\bytet#1{\hbox to 36pt{\hfill\.{#1}\hfill\vedgee}} % 34
\def\byteu#1{\hbox to 96pt{\hfill\.{#1}\hfill\vedgef}} % 84x32
\def\bytev#1{\hbox to 16pt{\hfill\.{#1}\hfill\vedge}} % 16x16
\def\bytew#1{\hbox to 16pt{\hfill\.{#1}\hfill\vedgec}} % 18
\def\bytex#1{\hbox to 17pt{\hfill\.{#1}\hfill\vedge}} % 16
\def\bytey#1{\hbox to 20pt{\hfill\.{#1}\hfill\vedge}} % 16
\def\bytez#1{\hbox to 18pt{\hfill\.{#1}\hfill\vedge}} % 16
\def\twobytes#1{\lower 5.5pt\vbox{\hbox to 28pt{\hfill\tick\hfill\tick} % 16
\hbox to 28pt{\hfill\.{#1}\hfill\untick}\hbox to 28pt{\hfill\tick\hfill\tick}}}
\def\twobytesa#1{\lower 5.5pt\vbox{\hbox to 30pt{\hfill\tick\hfill\tick} % 16
\hbox to 30pt{\hfill\.{#1}\hfill\untick}\hbox to 30pt{\hfill\tick\hfill\tick}}}
\def\twobytesb#1{\lower 10.5pt\vbox{\hbox to 30pt{\hfill\tick\hfill\tick} % weird
\hbox to 30pt{\hfill\.{#1}\hfill\unticka}\hbox to 30pt{\hfill\tick\hfill\tick}}}
\def\twobytesc#1{\lower 5.5pt\vbox{\hbox to 60pt{\hfill\tick\hfill\tick} % 16
\hbox to 60pt{\hfill\.{#1}\hfill\untick}\hbox to 60pt{\hfill\tick\hfill\tick}}}
\def\twobytesd#1{\lower 6.5pt\vbox{\hbox to 36pt{\hfill\tick\hfill\tick} % 18
\hbox to 36pt{\hfill\.{#1}\hfill\untickc}\hbox to 36pt{\hfill\tick\hfill\tick}}}
\def\twobytese#1{\lower 5.5pt\vbox{\hbox to 48pt{\hfill\tick\hfill\tick} % 16
\hbox to 48pt{\hfill\.{#1}\hfill\untick}\hbox to 48pt{\hfill\tick\hfill\tick}}}
\def\twobytesf#1{\lower 6.5pt\vbox{\hbox to 30pt{\hfill\tick\hfill\tick} % 18
\hbox to 30pt{\hfill\.{#1}\hfill\untickc}\hbox to 30pt{\hfill\tick\hfill\tick}}}
\def\twobytesg#1{\lower 3.5pt\vbox{\hbox to 24pt{\hfill\tick\hfill\tick} % 12
\hbox to 24pt{\hfill\.{#1}\hfill\untickd}\hbox to 24pt{\hfill\tick\hfill\tick}}}
\def\twobytesh#1{\lower 6.5pt\vbox{\hbox to 60pt{\hfill\tick\hfill\tick} % 18
\hbox to 60pt{\hfill\.{#1}\hfill\untickc}\hbox to 60pt{\hfill\tick\hfill\tick}}}
\def\twobytesi#1{\lower 6.5pt\vbox{\hbox to 50pt{\hfill\tick\hfill\tick} % 18
\hbox to 50pt{\hfill\.{#1}\hfill\untickc}\hbox to 50pt{\hfill\tick\hfill\tick}}}
\def\twobytesj#1{\lower 5.5pt\vbox{\hbox to 32pt{\hfill\tick\hfill\tick} % 16
\hbox to 32pt{\hfill\.{#1}\hfill\untick}\hbox to 32pt{\hfill\tick\hfill\tick}}}
\def\twobytesk#1{\lower 6.5pt\vbox{\hbox to 32pt{\hfill\tick\hfill\tick} % 18
\hbox to 32pt{\hfill\.{#1}\hfill\untickc}\hbox to 32pt{\hfill\tick\hfill\tick}}}
\def\twobytesl#1{\lower 5.5pt\vbox{\hbox to 40pt{\hfill\tick\hfill\tick} % 16
\hbox to 40pt{\hfill\.{#1}\hfill\untick}\hbox to 40pt{\hfill\tick\hfill\tick}}}
\def\threebytes#1{\lower 5.5pt\vbox{\hbox to 42pt{\hfil\tick\hfil\tick\hfil\tick}
\hbox to 42pt{\hfil\.{#1}\hfil\untick} % 16
\hbox to 42pt{\hfil\tick\hfil\tick\hfil\tick}}}
\def\threebytesa#1{\lower 5.5pt\vbox{\hbox to 45pt{\hfil\tick\hfil\tick\hfil\tick}
\hbox to 45pt{\hfil\.{#1}\hfil\untick} % 16
\hbox to 45pt{\hfil\tick\hfil\tick\hfil\tick}}}
\def\threebytesb#1{\lower 6.5pt\vbox{\hbox to 75pt{\hfil\tick\hfil\tick\hfil\tick}
\hbox to 75pt{\hfil\.{#1}\hfil\untickc} % 18
\hbox to 75pt{\hfil\tick\hfil\tick\hfil\tick}}}
\def\threebytesc#1{\lower 5.5pt\vbox{\hbox to 48pt{\hfil\tick\hfil\tick\hfil\tick}
\hbox to 48pt{\hfil\.{#1}\hfil\untick} % 16
\hbox to 48pt{\hfil\tick\hfil\tick\hfil\tick}}}
\def\fourbytes#1{\lower 6.5pt\vbox{\hbox to 72pt{\hfil\tick\hfil % 72x18
\tick\hfil\tick\hfil\tick}
\hbox to 72pt{\hfil\.{#1}\hfil\untickc}
\hbox to 72pt{\hfil\tick\hfil\tick\hfil\tick\hfil\tick}}}
%Next box same size as above but enclosure is to the right
\def\fourbytesa#1{\lower 6.5pt\vbox{\hbox to 72pt{\hfil\tick\hfil % 72x18
\tick\hfil\tick\hfil\tick}
\hbox to 72pt{\hfil\.{#1}\untickc}
\hbox to 72pt{\hfil\tick\hfil\tick\hfil\tick\hfil\tick}}}
\def\fourbytesb#1{\lower 5.5pt\vbox{\hbox to 64pt{\hfil\tick\hfil % 56x16
\tick\hfil\tick\hfil\tick}
\hbox to 64pt{\hfil\.{#1}\hfil\untick}
\hbox to 64pt{\hfil\tick\hfil\tick\hfil\tick\hfil\tick}}}
\def\fivebytes#1{\lower 5.5pt\vbox{\hbox to 70pt{\hfil\tick\hfil\tick\hfil
\tick\hfil\tick\hfil\tick} %16
\hbox to 70pt{\hfil\.{#1}\hfil\untick}
\hbox to 70pt{\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick}}}
\def\fivebytesa#1{\lower 5.5pt\vbox{\hbox to 75pt{\hfil\tick\hfil\tick\hfil
\tick\hfil\tick\hfil\tick} % 16
\hbox to 75pt{\hfil\.{#1}\hfil\untick}
\hbox to 75pt{\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick}}}
\def\fivebytesb#1{\lower 6.5pt\vbox{\hbox to 150pt{\hfil\tick\hfil\tick\hfil
\tick\hfil\tick\hfil\tick} % 18
\hbox to 150pt{\hfil\.{#1}\hfil\untickc}
\hbox to 150pt{\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick}}}
\def\fivebytesc#1{\lower 5.5pt\vbox{\hbox to 150pt{\hfil\tick\hfil\tick\hfil
\tick\hfil\tick\hfil\tick} % 16
\hbox to 150pt{\hfil\.{#1}\untick}
\hbox to 150pt{\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick}}}
\def\fivebytesd#1{\lower 5.5pt\vbox{\hbox to 80pt{\hfil\tick\hfil\tick\hfil
\tick\hfil\tick\hfil\tick} % 16
\hbox to 80pt{\hfil\.{#1}\untick}
\hbox to 80pt{\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick}}}
\def\fivebytesx#1{\lower 5.5pt\vbox{\hbox to 100pt{\hfil\tick\hfil\tick\hfil
\tick\hfil\tick\hfil\tick} % 16
\hbox to 100pt{\hfil\.{#1}\untick}
\hbox to 100pt{\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick}}}
\def\sixbytes#1{\lower 6.5pt\vbox{\hbox to 90pt{\hfil\tick\hfil\tick\hfil % 18
\tick\hfil\tick\hfil\tick\hfil\tick}
\hbox to 90pt{\hfil\.{#1}\hfil\untickc}
\hbox to 90pt{\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick}}}
\def\sixbytesa#1{\lower 3.5pt\vbox{\hbox to 72pt{\hfil\tick\hfil\tick\hfil % 12
\tick\hfil\tick\hfil\tick\hfil\tick}
\hbox to 72pt{\hfil\.{#1}\untickd}
\hbox to 72pt{\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick}}}
\def\sixbytesb#1{\lower 6.5pt\vbox{\hbox to 108pt{\hfil\tick\hfil\tick\hfil % 18
\tick\hfil\tick\hfil\tick\hfil\tick}
\hbox to 108pt{\hfil\.{#1}\hfil\untickc}
\hbox to 108pt{\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick}}}
\def\sixbytesc#1{\lower 6.5pt\vbox{\hbox to 150pt{\hfil\tick\hfil\tick\hfil % 18
\tick\hfil\tick\hfil\tick\hfil\tick}
\hbox to 150pt{\hfil\.{#1}\hfil\untickc}
\hbox to 150pt{\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick}}}
\def\sixbytesd#1{\lower 5.5pt\vbox{\hbox to 96pt{\hfil\tick\hfil\tick\hfil % 16
\tick\hfil\tick\hfil\tick\hfil\tick}
\hbox to 96pt{\hfil\.{#1}\hfil\untick}
\hbox to 96pt{\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick}}}
\def\sixbytese#1{\lower 6.5pt\vbox{\hbox to 96pt{\hfil\tick\hfil\tick\hfil % 18
\tick\hfil\tick\hfil\tici\hfil\tick}
\hbox to 96pt{\hfil\.{#1}\hfil\untickc}
\hbox to 96pt{\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick\hfil\tick}}}
\def\opensign#1{\lower 5.5pt\vbox to 16pt{\hbox to 10pt{\hfil\tick}\vfill
\hbox to 14pt{\hfil\.{#1}\hfil}\vfill\hbox to 14pt{\hfil\tick}}}
\def\textindentindent#1{\noindent\hbox to 40pt{\hss#1\ }\!}
\def\hanghang{\hangindent 60pt}
\def\textindentation#1{\noindent\hbox to 60pt{\hss#1\ }\!}
\def\hanging{\hangindent 40pt}
\def\hangang{\hangindent 30pt}
\def\textind#1{\noindent\hbox to 29pt{\hss#1\ ↑\!}
%\hangifdent29pt\textind{} (Case (iii) corresponds to the 

\defXtextinde#1{\noindent\hbox to 50pt{\hss#1\ @y8C|~)9IKMqQC]O≥w9QC9OSMI∃]h@jAai|~)9IKMq←giCIw9QE=qw9E¬gKYS9KgWS@b`aaPI1Ga%YKw9=I←i9
d~∃99←CYS≥]w9mMWS`@4b`aaQ⎇1mG∃]iKem9QE←aw8\Uy⎇9GexI⎇|~)9IKMqgaCG∃w9GQ¬dNh`↓|∩∃9⊃KM9E%ECw9ACe)A¬eS]I∃]h@beai9Q¬]OS]⊃K]h@Lqaiwy|~∃9⊃KM9]=aeKGm9SCi!eKYwqQgWS@gai9IKYm9skip -6.5pt\prec}}
\def\nopreceq{\mathrel{\hskip3pt\relv\hskip -6.5pt\preceq}}
\def\callseq#1{{\lineskip0pt\halign{\hbox to 113pt{\hfil##\quad}⊗\!
\save0\hbox{\hangindent 19pt\hbox par 235pt{\hangindent 19pt\strut ##\strut}}\!
\save1\hbox{\strut}\raise1ht1\hbox{\lower1ht0\box0}\cr #1}}}
\def\bracetext#1#2#3{\save0\vbox{\null\null}\!
\save1\vbox{\vskip#1ht0\vskip#1ht0\vskip-#2ht0\vskip-1ht0}\!
\save0\hbox{\raise.5ht1\hbox{$\left.\vcenter{\vskip#2ht0}\right\}$ #3}}\!
\vbox to 0pt{\vskip-1ht0\box0\vss}}
\def\lbracetext#1#2#3{\save0\vbox{\null\null}\!
\save1\vbox{\vskip#1ht0\vskip#1ht0\vskip-#2ht0\vskip-1ht0}\!
\save0\hbox{\raise.5ht1\hbox{#3 $\left\{\vcenter{\vskip#2ht0}\right.$}}\!
\vbox to 0pt{\vskip-1ht0\box0\vss}}
\def\bfn{\hbox{\bf n}}
\def\bfq{\hbox{\bf q}}
\def\bfk{\hbox{\bf k}}
\def\bft{\hbox{\bf t}}
\def\bfSigma{\hbox{\bf $\Sigma$}}
\def\bfz{\hbox{\bf 0}}
\def\bfL{\hbox{\bf L}}
\def\bfe{\hbox{\bf e}}
\def\ansalgstep #1 #2. {\anskip\noindent\hbox to 40pt{\!
\hbox to 19pt{\hss\bf#1\ }\hfil\bf#2. }\hangindent 40pt\!}
\def\bracex{\leaders\hrule height 1.5pt \hfill}
\def\dnbrace{$\char'772$\bracex$\char'775\char'774$\bracex$\char'773$}
\def\upbrace{$\char'774$\bracex$\char'773\char'772$\bracex$\char'775$}
\def\anscallseq#1{{\lineskip0pt\halign
{\hbox to 19pt{\hss\bf##\ }⊗\!
\hbox to 94pt{\hfil##\quad}⊗\!
\save0\hbox{\hangindent 19pt\hbox par 235pt{\hangindent 19pt\strut ##\strut}}\!
\save1\hbox{\strut}\raise1ht1\hbox{\lower1ht0\box0}\cr #1}}}
\def\mixansfive #1. #2{\def\\{\noalign{\penalty-200}}\annskip
\halign{\hbox to 19pt{##}⊗\rt{\it##}\quad⊗\lft{\tt##}\quad\!
⊗\lft{\tt##}\quad⊗\lft{\tt##}\quad⊗$\ctr{##}$⊗\quad\lft{\rm##}\cr
{\hfill\bf #1.\ }#2}}

\def\and{\chop to 0pt{\spose{\hskip-2pt\lower 5pt\hbox{\:C\char3}}}}
\def\aaalgstep #1. [#2]{\par\yskip\hangindent 29pt
	\noindent\hbox to 29pt{\hss\bf#1.\ }\hskip-.08em[#2]\xskip\!}
\def\phybox{\hbox{\vrule width 2pt height 6pt depth 2pt}}
\def\hide#1{\save 0\hbox{#1}\vbox to 0pt{\vskip-1ht0\box 0\vss}}

\def\+#1{\hbox to 0pt{\hskip 0pt minus 100pt #1\hskip 0pt minus 100pt}}
\def\rarrow{\relv\hskip -2.5pt\→} %|→ double right arrow
\def\zap{\mathrel{\hbox{$\vcenter{\hbox{$>$}\vskip-5pt\hbox{$~$}}$}}}
\def\anda{\mathrel{\hbox{$\vcenter{\hbox{$∧$}\vskip-7pt\hbox{$∧$}}$}}}
\def\ora{\mathrel{\hbox{$\vcenter{\hbox{$∨$}\vskip-7pt\hbox{$∨$}}$}}}
\def\naive{na\"\i ve}
\def\twoboxes#1{{\lineskip0pt\halign{
\save0\hbox{\hbox par 50pt{\strut ##\strut}}\!
\save1\hbox{\strut}\raise1ht1\hbox{\lower1ht0\box0}\quad⊗\!
\save0\hbox{\hbox par 275pt{\strut ##\strut}}\!
\save1\hbox{\strut}\raise1ht1\hbox{\lower1ht0\box0}\cr #1}}}
\def\Aslash{\hbox{$A$\hbox to 0pt{\hskip-9.5pt$\rslash$\hss}}}
\def\undertext#1{$\underline{\hbox{#1}}$}

\def\omod{\hbox to 25pt{\hfill$\vcenter{\hbox{\:@\char'140}}$\hfill}\hskip-25pt
\raise .3pt\hbox to 25pt{\hfill$\vcenter{\moveleft .2pt\hbox{\:emod}}$\hfill}}

\def\circle#1{\hbox to 10pt{#1\hskip-10ptminus10pt %use for math node single letters
\raise 6.944pt\hbox{\:@\char'141}\hskip0ptminus10pt}} %or double numerals
\def\circlea#1{\hbox to 4pt{#1\hskip-4ptminus4pt %use for single numerals
\raise 6.944pt\hbox{\:@\char'141}\hskip0ptminus4pt}}

\def\square#1{\save 0\hbox{#1}\!
\hbox to 1wd0{#1\hskip-1wd0\hss
\lower3.5pt\vbox{\baselineskip0pt\lineskip0pt\hrule
\hbox{\vrule height12pt\hskip12pt\vrule}\hrule}\hss
}}
\def\squarea#1{\save 0\hbox{#1}\!
\hbox to 1wd0{#1\hskip-1wd0\hss
\lower3.5pt\vbox{\baselineskip0pt\lineskip0pt\hrule
\hbox{\vrule height12pt\hskip28pt\vrule}\hrule}\hss
}}
\def\squareb#1{\save 0\hbox{#1}\!
\hbox to 1wd0{#1\hskip-1wd0\hss
\lower3.5pt\vbox{\baselineskip0pt\lineskip0pt\hrule
\hbox{\vrule height12pt\hskip16pt\vrule}\hrule}\hss
}}

\def\mixeight#1{\let\ttt=\tt\ninepoint\def\\{\noalign{\penalty-200}}
	\halign{\rt{\it##}\quad\tabskip0pt⊗\lft{\ttt##}\quad⊗\lft{\ttt##}\quad
	⊗\lft{\ttt##}\quad
	⊗\ctr{$ ##$}\qquad\tabskip0pt⊗\lft{\it##}\quad⊗\lft{\ttt##}\quad
	⊗\lft{\ttt##}\quad⊗\ctr{$ ##$}\cr#1\\}}
\def\mixnine#1{\let\ttt=\tt\ninepoint\def\\{\noalign{\penalty-200}}
	\halign{\rt{\it##}\quad\tabskip0pt⊗\lft{\ttt##}\quad⊗\lft{\ttt##}\quad
	⊗\lft{\ttt##}\qquad
	⊗\rt{\it##}\quad\tabskip0pt⊗\lft{\ttt##}\quad⊗\lft{\ttt##}\quad
	⊗\lft{\ttt##}\quad⊗\ctr{$ ##$}\quad⊗\lft{\rm##}\cr#1\\}}

\def\bytess#1{\hbox to 44pt{\quad\.{#1}\hfill\vedgec}} % 18

\def\threebytesd#1{\lower 6.5pt\vbox{\hbox to 48pt{\hfil\tick\hfil\tick\hfil\tick}
\hbox to 48pt{\hfil\.{#1}\hfil\untickc} % 18
\hbox to 48pt{\hfil\tick\hfil\tick\hfil\tick}}}

\def\bslash{\char'477 } % boldface slash 

\def\llangle{\left\langle\hskip-3pt\left\langle}
\def\rrangle{\right\rangle\hskip-3pt\right\rangle}
\def\mixfivea#1{\let\ttt=\tt\ninepoint\def\\{\noalign{\penalty-200}}
	\halign{\rt{\it##}\quad\tabskip0pt⊗\lft{\ttt##}\quad⊗\lft{\ttt##}\quad
	⊗\lft{\ttt##}\quad⊗\ctr{$ ##$}\quad⊗\rm##\hfil\cr#1\\}}
\def\ansalgstepa #1 #2. {\anskip\noindent\hbox to 45pt{\!
\hbox to 19pt{\hss\bf#1\ }\hfil\bf#2. }\hangindent 45pt\!}